Question

How many different 6-digit numbers can be formed using digits in the number 659942? How many of them are divisible by 4?

Answer 1

A 6-digit number is to be formed using digits of 659942, in which 9 repeats twice.

∴ Total number of arrangements

= `(6!)/(2!)`

= `(6 xx 5 xx 4 xx 3 xx 2!)/(2!)`

= 360

∴ 360 different 6-digit numbers can be formed.

For a number to be divisible by 4, the last two digits should be divisible by 4 i.e. 24, 52, 56, 64, 92, or 96.

Case I: When the last two digits are 24, 52, 56 or 64.

As the digit 9 repeats twice in the remaining four numbers, the number of arrangements 

= `(4!)/(2!)`

= `(4 xx 3 xx 2!)/(2!)`

= 12

∴ 6-digit numbers that are divisible by 4 so formed are 12 + 12 + 12 + 12 = 48.

Case II: When the last two digits are 92 or 96.

As each of the remaining four numbers is distinct, the number of arrangements = 4! = 24

∴ 6-digit numbers that are divisible by 4 so formed are 24 + 24 = 48.

∴ Required number of numbers framed = 48 + 48 = 96

Thus, 96 6-digit numbers can be formed that are divisible by 4.